Time frequency analysis and the bilinear Hilbert transform

The bilinear Hilbert transform is defined by

It has been shown recently that this bilinear operator

satisfies estimates of the type

for certain p, q, and r. This is the analogue of the

classical Riesz-Kolmogorow theorem for the (linear)

Hilbert transform. The purpose of this talk is to discuss

these and related results. A main tool is the decomposition of

functions into wave packets, i.e., well localized functions

whose Fourier transform is also well localized. Such decompositions

are frequently used in applied mathematics such as signal or

image processing.